Formal Semantics in Modern Type Theories:
Is It Model-theoretic, Proof-theoretic, or Both?⋆
Zhaohui Luo⋆⋆
Department of Computer Science
Royal Holloway, University of London
[email protected]
Abstract. In this talk, we contend that, for NLs, the divide between
model-theoretic semantics and proof-theoretic semantics has not been
well-understood. In particular, the formal semantics based on modern
type theories (MTTs) may be seen as both model-theoretic and prooftheoretic. To be more precise, it may be seen both ways in the sense
that the NL semantics can first be represented in an MTT in a modeltheoretic way and then the semantic representations can be understood
inferentially in a proof-theoretic way. Considered in this way, MTTs
arguably have unique advantages when employed for formal semantics.
1
Introduction
In logic, there are two well-developed traditions of semantic theories: modeltheoretic semantics and proof-theoretic semantics. The former has been developed in Tarski’s tradition while the latter by the logicians Gentzen and Prawitz
and the philosopher Dummett among others (see, for example, [17]).
In formal semantics of NLs, the model-theoretic tradition has been dominant
since the seminal work by Montague, a student of Tarski, who has used Church’s
simple type theory as an intermediate logical language for model-theoretic semantics [27]. The proof-theoretic semantics for NLs, however, has not been well
developed, although there is recent work by Francez and colleagues in that direction (see, for example, [13,14]).1
We contend that, for formal semantics of NLs, the divide between modeltheoretic semantics and proof-theoretic semantics has not been well-understood
⋆
⋆⋆
1
This paper is associated with the invited talk of mine in Logical Aspects of Computational Linguistics 2014. Its published version in LACL proceedings (LNCS 8535)
contains some typos which have been corrected here.
This work is partially supported by the research grant F/07-537/AJ of the Leverhulme Trust in U.K.
In philosophy, different kinds of philosophical semantics have been proposed, including the conceptual role semantics such as semantic inferentialism that has been
advocated by Brandom [4,5] and others. Note that such philosophical semantic studies have rather different assumptions and more ambitious objectives as compared to
formal semantics, although they have interesting influences on the latter.
and may have been over-exaggerated. In fact, the formal semantics in modern
type theories (MTT-semantics, for short) [32,22] may be seen as both modeltheoretic and proof-theoretic. To be more precise, it may be seen both ways in
the following sense: the NL semantics can first be represented in an MTT in a
model-theoretic way and then the semantic representations can be understood
inferentially in a proof-theoretic way.
The MTT-semantics is in the tradition of the Montague semantics [27] from
which, however, it has a key difference which may be summarised as follows:
– In the Montague semantics, the meanings are given in the set-theoretical
models of the simple type theory, which acts as the intermediate logical
language for model-theoretic semantics.
– In the MTT-semantics, the meanings are given in the language MTT itself,
not in the ‘models’ of the MTT.
Put in another way, in an MTT-semantics, the MTT is not acting as an intermediate language, but rather as the meaning-carrying language itself.2 On the
one hand, the MTT-semantics may be seen as model-theoretic because, in such
semantics, an MTT is employed as a representational language and it can do so
because of its rich representational structure3 as well as its internal logic. On
the other hand, the MTT-semantics may be seen as proof-theoretic because the
meanings of MTT-judgements can be understood by means of their inferential
roles according to the meaning theories such as that developed by Martin-Löf
for his type theory [25,28].
Besides this main theme, some related technical developments are to be reported. We describe a method of representing situations (incomplete possible
worlds), which may be infinite or involve more sophisticated phenomena, by
means of finite contexts in type theory with manifest entries (c.f., the study
of manifest fields in [20]) and/or subtyping entries (c.f., the study of coercion
contexts and local coercions in [22,23]) as well as the usual membership entries.
The extension with such contextual entries has nice meta-theoretic properties.
Extending a brief argument in §1.2.3 of [18], in agreement with Dummett
[12], I shall argue that a proof-theoretic semantics of MTTs should take both
aspects of inferential use (verification and consequential application) seriously
and that this applies to the understanding of judgements and type constructors
in MTTs.
In this talk, after giving a summary of the MTT-semantics,4 I shall elaborate
the above theme and point out that, if we accept such views, MTTs may have
some unique advantages when employed for formal semantics of NLs.
2
3
4
It could be possible for one to consider an MTT as an intermediate language for
model-theoretic semantics, although this view is not usually taken and it might be
difficult as well. See §4 for a brief discussion in this respect.
To see that MTTs provide rich representational structures, one may compare the
notion of type with that of set and the notion of an MTT context with that of
situations or incomplete possible worlds. These will be elaborated below in §2.
The summary is omitted in the current paper (but see §2.1). For its references, see
[32,22,21,9] among others.
2
2
Model-theoretic Characteristics of MTT-semantics
A modern type theory (MTT) has a rich type structure, much richer than the
simple arrow types as found in the Montagovian setting.5 The richness of type
structures in an MTT allows it to be used as a representational language for
formal semantics. Besides logical propositions (i.e., types that are understood
as propositions), it contains many types which are employed for various useful representational purposes in constructing a formal semantics. These types
include those interpreting (to mention a few examples) logical operations, common nouns, adjectival modifications, adverbial modifications, coordination, and
also more advanced features such as subtyping, copredication and linguistic coercions.
In an MTT, the notion of judgement is the most basic. A typical form of
judgement is:
Γ ⊢ a : A,
which intuitively says that ‘object a is of type A in context Γ ’, where a context,
in its traditional form,6 is a finite sequence of variable-type pairs of the form
x1 : A1 , x2 : A[x1 ], ..., xn : An [x1 , ..., xn−1 ],
by which we assume that xi be of type Ai [x1 , ..., xi−1 ] in which the ‘previous’
x1 , ..., xi− 1 may occur (i = 1, ..., n).
Employing MTTs for formal semantics, the following two points are worth
being made explicit:
1. The types in MTTs can be used to represent collections of objects (or just
called sets, informally) in a model-theoretic sense, although they are syntactic entities in MTTs.
2. The contexts in MTTs can be used to represent situations or (incomplete)
possible worlds.
In this section, we shall illustrate these two points. First, in §2.1, we briefly
describe how various type constructors in MTTs can be used in representing
linguistic features in formal semantics. We then, in §2.2, give a simple example
to explain how MTT contexts can be used to represent situations and, in §2.3,
show how more sophisticated situations like infinite ones can be represented in
finite contexts by means of manifest entries (c.f., manifest fields in [20]) and
5
6
Technically, MTTs may be classified into predicative type theories such as MartinLöfs type theory [25,28] and impredicative type theories such as the calculus of
constructions (CC) [11] and the unifying theory of dependent types (UTT) [18]. In
computer science, MTTs have been implemented in the proof assistants such as Coq
[10] which has been used to implement the MTT-semantics and conduct experiments
in Natural Language Inference [9]. In this paper, the type theory UTT [18] is used
when concrete examples are given.
We shall later in the paper extend the notion of context with manifest entries and
subtyping entries.
3
subtyping entries (c.f., the study of coercion contexts and local coercions in
[22,23]).7
2.1
Types as Sets: Examples
The rich type structure in MTTs provides adequate and useful tools for semanticists to construct formal semantics. Here, we only mention several examples
without going into their details, because most of them have been explained in
details in other places, and the appropriate references are given in each case.
1. Dependent sum types (Σ-types Σ(A, B) which have product types A × B as
special cases).
– For example, Σ-types have been used to interpret adjectival modifications when the adjectives are intersective and subsective [32,8]. Note
that subtyping is essential for such Σ-type interpretations to be adequate [22].8
2. Dependent product types (Π-types Π(A, B), which have arrow-types A →
B as special cases). These are basic dependent types that, together with
universes (see below), provide polymorphism among other things.
– For instance, verb modifying adverbs are typed by means of dependent
Π-types (together with the universe cn of common nouns) [22,6].
3. Disjoint union types (A + B).
– For instance, disjoint union types have been proposed to give interpretations of privative adjectives [8].
4. Universes. These are types of types, very useful since they make it possible
to treat various collections of types as internal totalities. Typical examples
of universes in MTT-semantics include, among others,
– P rop, the universe of logical propositions, as found in impredicative type
theories such as CC and UTT;
– cn, the universe of (the interpretations of) common nouns [22]; and
– LT ype, the universe of ‘linguistic types’, introduced in studying coordination [7].
5. Dot-types (A • B). These are special types introduced to study the linguistic
phenomena of copredication [22]. It is always worth mentioning that coercive
subtyping is essentially employed in the formulation of the dot-types.
Besides the above, I should also emphasise that subtyping is crucial for an
MTT to be a viable language for formal semantics. Furthermore, also very importantly, subtyping is needed when considering many linguistic features such as
7
8
It is worth remarking that, formally, the studies in [20] and [22,23] are technically
more complex in both cases: we studied manifest fields (not just manifest contextual entries) in [20] and local coercions (not just subtyping contexts) in [22,23]. In
this paper, when we only consider such entries in signatures (see the brief formal
treatment of the extension at the end of §2.3), it is formally simpler.
See §3.3 of [32] for a very interesting discussion of the problem of ‘multiple categorisation of verbs’. A satisfactory solution is an adequate subtyping mechanism for
MTTs, provided by the framework of coercive subtyping [19,24].
4
copredication [1,31]. However, introducing subtyping into a Montagovian semantics has proven difficult, if not impossible. Coercive subtyping in MTTs solves
this problem – it not only provides a satisfactory solution to the copredication
problem [22] but a useful way to formalise various linguistic coercions [2].
2.2
Situations Represented as Contexts: a Simple Example
Here, we present a small example to show how an MTT can be used as a representational language. Situations,9 or incomplete possible worlds, can be represented
in MTTs as contexts. This was first studied by Ranta in [32]. Our example is
simple; for instance, the domain of the situation in the example is finite. This
will serve as a basis for our discussions on how more sophisticated situations such
as those involving infinity or other special circumstances should be represented
in §2.3, where we will show how manifest entries and subtyping entries can be
used for such purposes.
Example 1. The example, taken from Chapter 10 of [33], is about an (imagined) situation in the Cavern Club at Liverpool in 1962 where the Beatles were
rehearsing for a performance. This situation can be represented as follows.
1. The domain of the situation consists of several peoples including the Beatles
(John, Paul, George and Ringo), their manager (Brian) and a fan (Bob). In
type theory, this can be represented be means of the following context Γ1 :
Γ1 ≡ D : T ype,
John : D, P aul : D, George : D, Ringo : D, Brian : D, Bob : D
2. The assignment function which, for example, assigns predicate symbols such
as B and G to the propositional functions expressing ‘was a Beatle’ and
‘played guitar’, respectively. We can introduce the following in our context
to represent such an assignment function:
Γ2 ≡ B : D → P rop, bJ : B(John), ..., bB : ¬B(Brian), b′B : ¬B(Bob),
G : D → P rop, gJ : G(John), ..., gG : ¬G(Ringo), ...
There are other predicate symbols as well: for example, there may be M
being assigned the propositional function expressing ‘was a manager’, etc.
We omit them here.
Eventually, we obtain a context Γ ≡ Γ1 , Γ2 , ..., Γn that represents the situation.
We shall then have, for instance,10
Γ ⊢ G(John) true and Γ ⊢ ¬B(Bob) true.
where G(John) and B(Bob) are the semantic interpretations of John played Guitar
and Bob was a Beatle, respectively.
9
10
Here, of course, we use the notion of situation informally, not in the formal sense in
Situation Semantics [3].
A judgement Γ ⊢ A true means that Γ ⊢ a : A for some a.
5
As we see from the above example, contexts in an MTT can be employed to
represent situations model-theoretically and, in this endeavour, type theory is
used as a meaning-carrying language (like set theory) rather than a language to
be further interpreted.
2.3
Representing More Sophisticated Situations: Manifest and
Subtyping Entries
The above example of situation is extremely simple: in particular, its domain D
is finite. In general, one usually considers more sophisticated situations in which,
for example, the domain is infinite or there are some other more sophisticated
phenomena. In such cases, only using the traditional notion of context, which
has only membership entries of the form x : A, is not enough. Here we consider
two extensions to the notion of context:
1. Manifest entries of the form x ∼ a : A, where a : A.11
2. Subtyping entries of the form c : A < B, where c is a functional operation
from A to B.
They will provide extra powers in representing situations as contexts.
Manifest Entries in Contexts. A manifest entry in a context is of the form
x ∼ a : A,
(1)
Informally, it assumes that x behave exactly the same as a of type A. Put in
another way, in any place that we could use an object of type A, we could use x
which actually plays the role of a. Formally, the above manifest entry (1) is an
abbreviation of the following entry:
x : 1A (a),
where 1A (a) is the inductive unit type parameterised by A : T ype and a : A,
whose only object is ∗A (a). We also assume that 1A (a) <ξA,a A, where ξA,a (x) =
a for every x : A.12 We have the following derivable formation rule for manifest
contextual entries:
Γ ⊢a:A
Γ, x ∼ a : A valid
(x ̸∈ F V (Γ ))
Using manifest entries, we can not only simplify contextual representations
(the following Example 2) but represent infinite or other sophisticated situations
by means of the finite contexts in type theory.
11
12
Manifest entries are different from the definitional entries of the form x = a : A
in proof assistants – the former is just a notational abbreviation in the framework
of coercive subtyping (see below), while the latter involves real extensions whose
metatheory has proved to be not easy. For a study of the latter for Pure Type
Systems, see [34].
We omit the formal definitions of 1A (a) and the formal details for the coercion ξ
here; see [20].
6
Example 2. With manifest entries, the situation in Example 1 can be represented
as the following context:
D ∼ aD : T ype, B ∼ aB : D → P rop, G ∼ aG : D → P rop, ... ...
(2)
where
– aD = {John, P aul, George, Ringo, Brian, Bob} is the finite type (not
the finite set) consisting of John etc.
– aB : D → P rop, the predicate ‘was a Beatle’, is an inductively defined
function such that aB (John) = aB (P aul) = aB (George) = aB (Ringo) =
T rue and aB (Brian) = aB (Bob) = F alse.
– aG : D → P rop, the predicate ‘played guitar’, is an inductively defined
function such that aG (John) = aG (P aul) = aG (George) = T rue and
aG (Ringo) = aG (Brian) = aG (Bob) = F alse.
In other words, Γ1 in Example 1 is now expressed by the first entry of (2) and
Γ2 in Example 1 by the second and third entries of (2).
Also, the domain in the situation in Example 1 is finite and, therefore, we
can use the finitely many membership entries as in Example 1 to represent it.
However, sometimes we have to deal with infinite domains. For instance, it is
not difficult to imagine a situation where domain D consists of infinitely many
things that we may represent by the natural numbers and some predicates are
represented as propositional functions such as f over D. In such a case, we can
use manifest entries like
D ∼ N at : T ype, f ∼ f-defn : D → P rop
to introduce the infinite domain D and the infinite propositional function f ,
where f-defn can be defined inductively (in this case by inductive definition over
the inductive type N at).
Subtyping Entries in Contexts. Another useful form of contextual entries is the
subtyping entries, for coercive subtyping, which are called coercion contexts in
[22,23]. Formally, a subtyping entry is of the following form:
c : A < B,
where A and B are types and c is a functional operation from A to B. In a
context with such an entry, A is a subtype of B via. coercion c.
Subtyping plays an essential role in MTT-semantics. It is not difficult to see
that subtyping contextual entries are very useful in representing situations as
contexts. As a simple example, one may introduce a subtyping entry to assume
that everything in the domain D in Example 2 is a human: D < Human.
Manifest entries and subtyping entries are useful for some sophisticated circumstances as well. For instance,13 one may imagine a situation (e.g., in the snake
13
Thanks to Nicholas Asher for this example (in a private communication).
7
exhibition of a zoo) where every animal is a snake and such a situation may be
represented by a context with the manifest entry Animal1 ∼ Snake : T ype or a
subtyping entry expressing that Animal1 is a subtype of Snake, where Animal1
stands for the type of animals in the particular situation. One may regard such
entries as unusual, although they describe the situation faithfully.
Formal Treatment of the Contextual Extensions. The above extension of the
notion of contexts with manifest and subtyping entries can formally be considered by introducing the notion of signature. Informally, signatures can be used
to represent situations in formal semantics. Formally, we consider the following
forms of judgements:
Γ ⊢Σ J
(3)
where Γ is a context in the traditional sense (a sequence of membership entries
of the form x : A), J is one of a : A, a = b : A, A type or A = B type, and Σ is
a signature, a sequence of entries of one of the following forms:
1. x : A (usual membership entries)
2. x ∼ a : A (manifest entries)
3. c : A < B (subtyping entries)
What is new here is signatures, which could formally be regarded just as initial
contextual segments whose entries will never be abstracted to the right of the
⊢-sign. In other words, the judgement (3) could be treated as Σ, Γ ⊢ J except
that the entries in Σ cannot be abstracted.14
Since manifest entries of the form x ∼ a : A are just abbreviations of membership entries of the form x : 1A (a) (together with coercion ξA,a ), it is straightforward to see that the extended type theory preserves the nice properties of
the original type system as long as the signatures concerned are coherent15 . For
instance, if Σ is coherent and Γ ⊢Σ a : A, then all of the terms a, A, and those
in Γ and Σ are strongly normalising (and, as a consequence among many others,
the embedded logic is logically consistent).
3
Proof-theoretic Characteristics of MTT-semantics
Besides model-theoretic semantics, there is another kind of formal semantics:
proof-theoretic semantics. Proof-theoretic semantics was pioneered by Gentzen
[15], developed by Prawitz [29,30] and Martin-Löf [25,26] (among other logicians)
to study meaning theories of logical systems, and studied by Dummett [12]
and Brandom [4,5] (among other philosophers) to study general philosophical
theories of meaning of NLs. Proof-theoretic semantics is a form of inferential
semantics in that it not only takes inference seriously but regards it as the central
14
15
This is similar to the treatment of signatures in Edinburgh LF [16].
If a well-formed signature Σ is coherent then Γ ⊢Σ A <c B and Γ ⊢Σ A <c′ B
imply Γ ⊢Σ c = c′ : (A)B. We do not get into the formal details here; see [19].
8
concept of meaning theories and such a direct link to inference is regarded as a
key advantage of proof-theoretic semantics.16
3.1
Proof-Theoretic Semantics of MTTs
The MTT-semantics does not only have model-theoretic characteristics, as studied in the above section, but proof-theoretic characteristics. This is because that
the meanings of judgements, which are the basic sentences in MTTs, can be
understood in a proof-theoretic meaning theory. Martin-Löf has carried out a
whole programme of proof-theoretic semantics, studying and developing it for
his type theory [25,26].17 Therefore, once MTT-semantics has been constructed,
the statements in the semantics (i.e., judgement in the MTT) can be understood
proof-theoretically.
Studying proof-theoretic meaning theories, people have considered two aspects of use: verification and consequential application. For MTTs, the first
aspect is to consider how a judgement can be correctly asserted while the other
(second) aspect is to consider what consequences it has to accept that a judgement is correct. The verificationist thinks that the first aspect of verification
is the (possibly only) central conceptual focus in meaning theories while the
pragmatist thinks that the second aspect of consequential application is instead
central. For example, under the verificationist view on meaning explanations for
type theory, one may consider such a meaning explanation of the judgement
a : A (I omit the context Γ here): it can correctly asserted if a computes to a
canonical object of type A.
Personally, I take the view that both aspects of use are essential for a meaning
theory of MTTs. This point of view is in agreement with Dummett [12] (if I read
him correctly) and was briefly discussed in §1.2.3 of [18].18 More concretely, this
view says that the meanings of the type constructors are given by both of their introduction and elimination rules. Accordingly, the meanings of MTT-constructs
and, in particular, the MTT-judgements, are given proof-theoretically (or inferentially, in more general terms). When the type constructors and judgements
are employed in formal semantics, such proof-theoretic meanings can be taken
as the basis of the semantics.
16
17
18
There are philosophical arguments that conceptual role semantics (eg, prooftheoretic semantics) supports the view in cognitive science that something is meaningful just because that it plays a certain role (eg, inferential role) in a person’s
psychology. I ignore this because here proof-theoretic semantics is considered as an
approach to formal semantics.
But see Conclusion for the remarks on future work on meaning theories on MTTs
in general and some existing problems (for example, about meaning explanations of
hypothetical judgements).
A reason for taking this view is that an MTT is much more sophisticated than
first-order logical operators. However, I cannot detail its arguments in this paper.
9
Γ, j isa X ⊢ S[j]
Γ ⊢ S[every X]
(eI)
(eE)
Γ ⊢ S[every X]
Γ ⊢ j isa X
Γ ⊢ S′
(j fresh)
Γ, S[j] ⊢ S ′
Γ ⊢ j isa X Γ ⊢ S[j]
Γ ⊢ S[some X]
(sI)
(sE)
Γ ⊢ S[some X]
Γ, j isa X, S[j] ⊢ S ′
Γ ⊢ S′
(j fresh)
Fig. 1. The PTS-rules for a basic language in [13].
3.2
Francez’s Proof-Theoretic Semantics
In the literature, there have been some, but not many, researches on prooftheoretic semantics of NLs. The most notable is the recent work by Francez and
his colleagues in a series of papers (see [13,14] among others). What I shall do
here is to embed the basic rules of Francez’s proof-theoretic semantics in [13] into
the MTT-semantics and show that they are actually derivable or admissible.19
For instance, consider the rules of Figure 1 in [13], repeated here in our
Figure 1 (we omit the structural rules and its treatment of scope here, which is
irrelevant to our point and can be restored if needed). Employing the following
‘translation’ [[ ]], it is easy to see that the rules in Figure 1 are all derivable or
admissible in the MTT semantics:
– The syntactic isa corresponds to the colon sign (:) in type theory. Thus, for
example, [[j isa X]] = j : [[X]].
– [[every X]] = ∀([[X]]) of type ([[X]])P rop. For instance, the sentence (4) is
interpreted as (5) after translation.
(4) Every man walks.
(5) ∀([[man]], [[walk]])
– [[some X]] = ∃([[X]]) of type ([[X]])P rop. (Example omitted.)
For instance, the rule (eI) becomes
(eI ′ )
[[Γ ]], j : [[X]] ⊢ [[S[j]]]
[[Γ ]] ⊢ [[S[∀(X)]]]
which is admissible according to ∀-introduction. (Note: the condition that j is
fresh is now embodied in the well-formedness of [[Γ ]], j : [[X]].)
19
I cannot make serious comparisons here and further work is needed to compare these
two approaches in a more exact way.
10
4
Concluding Remarks
In this talk, I have argued that the MTT-semantics may be seen as both modeltheoretic and proof-theoretic, with situations represented as contexts and judgements understood inferentially.
There are several interesting directions for future work, either proof theoretically or model theoretically. Proof theoretically, it is important to develop
further the meaning theories of modern type theories in general. For instance, it
is interesting to study and obtain a better meaning explanation of an impredicative universe like that found in UTT. Even just for predicative type theories like
that of Martin-Löf, people may argue that it is still unclear how hypothetical
judgements should be explained satisfactorily. Further studies are called for.
As remarked in Footnote 2, one might consider model theory of MTTs, thinking of which as intermediate languages to be further interpreted when used for
formal semantics. However, it is unclear whether such an approach can be successful, partly because that there is no such a general notion of models for MTTs
as yet, although some initial work on models of generalised algebraic theories
have been done. Substantial work is needed to get this clarified and developed.
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